Chapter 1"However, the type of learning that is going on as a result looks so different from the kinds of learning described by most educational theorists that it is essentially invisible."

I think this quote addresses one of the major themes for this chapter and the purpose for the EDUC 530 course. Throughout this book and course we have been introduced to different technologies and tools that we can use to leverage the massive shift in information availability that has occurred in our society. The traditional view of the teacher being the only true font of knowledge has been broken in the information age. Any individual who is motivated can find and learn any topic they wish online. The only thing they would lack is a ready guide to direct their energy and enthusiasm for a topic.

In the new culture of learning that has been created in our modern information age we need to reassess the role of the classroom teacher. Teachers need to recognize that students can gain knowledge from multiple sources and have tools available that makes previous methods and knowledge obsolete. Getting students interested in our individual subjects and showing them how to play and learn with the tools and skills provided would serve our students better than attempting to shoehorn them into learning under an educational ideology from a previous century.

Educators need to help students learn how to leverage and use these new tools to effectively become personal learners. Students need to understand that they have the tools and ability to find solutions to their own questions and shouldn't be constantly reliant on the gift of knowledge and answers from others.

Chapter 2Concerning the model of education used throughout the early 20th century.

"This model, however, just can’t keep up with the rapid rate of change in the twenty-first century. It’s time to shift our thinking from the old model of teaching to a new model of learning."

Considering the new tools and methods used by individuals and institutions to disseminate information and share ideas we have to revise our ideas about education. This chapter asks the reader to re-contextualize the role and structure of formal education. Given that students have a wealth of knowledge at their disposal we should no longer consider the teacher to be the only source of information in the classroom.

A better view would hold that the teacher is the best guide in the classroom. The teacher has a better understand of the material being covered and has insights into possible problems and questions that can come up concerning the material. The teacher can help to direct students away from false information and help to structure the discussion so that students make progress in understanding and accessing the information. The teacher is a conductor directing the learning.

A key point that the authors make is that in the new culture of learning questions are more highly valued and appreciated. Questions can show that a student is interested in the material and is trying to draw connections and correlations to previous knowledge. Questions can lead to deeper appreciation of material and methods. Questions show that a student is trying to understand the material at a deeper level and cares about it.

Chapter 3"Many educators, for example, consider the principle underlying the adage, “Give a man a fish and feed him for a day, teach a man to fish and feed him for a lifetime,” to represent the height of educational practice today. Yet it is hardly cutting edge. It assumes that there will always be an endless supply of fish to catch and that the techniques for catching them will last a lifetime"

This chapter calls for educators to re-evaluate the nature of knowledge as we traditionally view it. The authors illustrate their idea by highlighting how Wikipedia is fundamentally a more sound system of collecting knowledge than traditional encyclopedias. They don't argue that Wikipedia is more correct or rigorous than traditional paper encyclopedias, they argue that it is more transparent and thus more adaptive. The reader can source information and track changes to articles. Reading the history of an article can show multiple views and ideas about a topic that were never apparent when reading an entry in a paper document.

This view can be expanded to look more facets of knowledge and content distribution. Students can freely explore and extend their understanding about a subject if they desire to. They can research any topic or material that is covered in the standard K-12 curriculum. That they don't freely choose to explore this wealth of knowledge available to them points out that teacher are failing to instill in students an understanding of the value and power of their subject matter.

This vast store of information and opportunity that is available to students is being ignored because it is being packaged in way that is ill suited to education in our current age and fails to compete with topics that students are placing value and time into.

Wagner lists the following seven skills as being essential for the success of students in the future. While I might have labeled the skills and abilities he choose differently, I feel that he is correct in identifying these as critical skills students will need in their future.

1st: Critical Thinking and Problem SolvingThis is probably the most useful skill anyone can have in general and is the most showing when it's missing. The ability to critically analyze a problem and identify workable solutions is critical for success with half of the skills list below. Students need to understand how to view problems and break them down into manageable and solvable pieces. Bundled in with this students need to have the patience and dedication to see a project through and not quit or give up if the solution isn't simple or quick.

2nd: Collaboration Across Networks and Leading by InfluenceStudents have to learn to work together and not in isolation. Few professions in the world have employees working by themselves on problems and testing and training students for this type of non-existent job is a disservice to the student. Students are better prepared for life by understanding how to work with a group to collaboratively find and implement solutions to problems. Students need to learn how to organize and direct themselves without constant supervision if they are to succeed in the world.

3rd: Agility and AdaptabilityI feel that this could have been partially bundled into Critical Thinking. Students need to understand the systems and problems they are working on so that they can see how they can implement solutions and identify problems. Agility can only come when you understand what you are working on and what methods are applicable. Adaptability means that a student can adapt to changes in the nature or their work or the problem being worked on. Students can't be stuck in a mindset that one method or solution will be correct for everyone problem and situation for the rest of their lives. They need to understand what skills they have and how they can be adapted to new situations and tasks.

4th: Initiative and EntrepreneurialismStudents have to view themselves as more than just cogs in a system to survive in the future. They need to understand that the job security and complacency that was acceptable is no longer the norm. Students need to take control of their lives and their output going forward. They need to learn how to take initiative in finding solutions to problems they have identified and being leaders for change when needed.

5th: Effective Oral and Written CommunicationI feel that this skill has overlap with number 2. Students need to learn how to effectively share and communicate their ideas with others. If we are to help prepare students for jobs where they will be expected to work with a team to create and implement solutions they need to have an understanding of how to present their ideas and opinions in a logical and coherent fashion so as to persuade others.

6th: Accessing and Analyzing InformationWith the constant access to information through the internet and connected devices students need to understand how to critically access the information they find online and receive from secondary sources. Students should be able to validate statements they hear and read, and know how to present evidence either for or against a given view.

7th: Curiosity and ImaginationAre both necessary yet ambiguous skills to have or demonstrate. Teaching a student to have imagination is a matter of nurturing their imagination as we have to show the student that their ideas and views have worth and provide insight. Students will never display their imagination or creativity if they don't feel safe expressing their ideas. Similarly curiosity can only come if we show students that their questions have worth. If students feel that they are making worthwhile connections and contributions with their questions they are more likely to explore deeper into subjects and ideas. Students need some kind of hook if they are going to have a desire to dig deeper into a subject and educators need to ensure they don't squash that desire.

Extra SkillsI feel that preservation could have been included in the list more heavily as it's own specific skill. The ability to stick with a hard problem and not quit is needed in every profession and should be more highly valued. Students need to be exposed to truly difficult problems more often in their lives so that they understand how much effort is required to find a solution. Real problems are never set up like word problems and the tests life throws at people rarely call for a single obscure fact. Students need to learn to identify problems in their world and create and carry out solutions if they are going to have success in the future.

I also feel that their should have been a skill focused on modern technological usage. While Wagner touches on technology in Collaboration Across Networks and Assessing and Analyzing Information, students need to have a fundamental understanding of how the can find and use technologies available in their lives to solve problems. Students also need to understand how they can identify problems that don't currently have an available tool or solution. Identifying which tool or process is best for finding a solution is an actual job skill that students are not currently being trained to meet.

What can I commit to in my classroom this year?Currently I'm teaching Business Math to Seniors and I hope that I am achieving some of these goals with my instruction. Students have collaborated with partners and groups to design budgets for business plans and moving out of their parents homes. Students learned about State and Federal taxes and completed a mock Federal Income Tax return online. We recently had a Financial Adviser as a guest speaker talk to the students about his job and what services a Financial Adviser provides to clients.

I hope that a lot of these lessons and activities have a greater connection to the students than a worksheet of solving for x. I feel that the material being covered is interesting to them considering that it has value in their near future and isn't created or modeled in a vacuum. Students are making choices and decisions that they will have to make throughout their lives in my class.

How will you measure success?With student involvement mostly. I can't dictate how grading and curriculum is structured at my current site but I feel that if the students are engaging with the material and are asking relevant questions, and are actively trying to link the material to their own lives, that this material will have a longer lasting impact on their lives. In the future I would like to implement portfolio based assessments like High Tech High. I feel that their approach to accessing students understanding is more inline with the experience that students will have outside of school in their actual lives. Preparing them for the world as it actually exists is of greater value to me than preparing them for a world where they individually answer mathematical problems given a specific time constraint.

In reading these two chapters I ran into a number of themes and ideas that are regularly expressed throughout the credential program.

Chapter 3:There is a real need for students to have a fundamental understanding of the material that can not be properly accessed or viewed through a traditional multiple choice exam structure. The heavy movement towards standardized tests and exit examinations has created an environment where teachers and students feel pressured to only study and learn specific topics that are tested. I have seen students in my class that can mechanically solve algebraic problems for x, but when asked have no clear understanding of why they are doing the operations. They had the process explained to them once and then repeated the process 30 more times on a worksheet for homework. They have been practicing problem solving using this method for so long now that students can't even solve simple problems that aren't clearly structured to fit into an equation.

Chapter 4:This chapter covers the need for educators to truly treat their career as a profession. Many teachers have accepted a mindset that their current work and efforts are enough. They meet the requirements for professional development mandated by the district and they go through the annual review by an administrator and that is it. True professionals should always be striving to improve and better their work. They should consider their jobs to be of worth and actually be angry about other teachers not performing at the expected level.

It's not that teachers should be jealous that a different department has easier state standards to meet. It's that some teachers are actively teaching badly and everyone accepts that it is more trouble to fix a bad tenured teacher than to just ignore the problem.

This chapter highlights the issues Wagner has seen and identified during his time as a teacher and administrator. Teachers need to actually treat their profession as a profession and not give it lip service. Settling for the status quo isn't fair to the students or society and teachers need to stop using it as a crutch for not improving their practice. If it's too much effort and trouble to teach relavent information then maybe teaching shouldn't be someone career.

In the military they wont let a non-performer re-enlist. While some people scrap by under the radar, the system is designed so that if you are not advancing and becoming more competent and skilled you are not given the option to stay on for another 4 years.

Chapter 5 deals with issues that are regularly covered in our Math Methods course. A distinct problem with Math education over the last several decades has been the push to teach only the material covered by the standardized tests. The interest and appreciation of math that professionals find useful everyday isn't being translated down to students.

Modern mathematics classes need to create more dynamic and engaging lessons that students can find personal worth in. If a student can't find an interesting reason or justification for the work they are doing in a classroom they will forget about the material as soon as they can.

Wagner highlights a lot of these issues and shows why modern students find a disconnect with their education. The methods being used to teach them were created in an previous industrial period from their perspective and fail to leverage any of the skills or tools that students have available to them currently. The requirement that students have to rote memorize most historical facts is archaic in a world with constant internet. Having to hand calculate values that are best left to a computer is rightly seen as tedious work. Testing students on their ability to perform either of these tasks fails to assess their ability to succeed in the world they currently live in.

To correct these issues Wagner points out areas where instructors can change and redirect education to more closely meet the needs and expectations of students. Learning should leverage multimedia and connection more often, students can stop, rewind, and replay videos covering many of the topics we are already trying to cover in class. Class time could be better utilized having students discuss what they observed and thought about the video than spent listening to lecture. Learning should happen as a discovery process, students will find more value in the information if they create the links and connections between different topics and discover the shortcuts themselves. Learning should include creation, students need to generate their own artifacts and memories to hook into their learning. The subject and material needs to be memorable if we are going to expect students to hold on to it for more than a week.

Chapter 6 deals with schools that work, or more specifically schools that are actually meeting the needs of their students and properly preparing them for the world outside of school. The school that is most prominently featured is High Tech High in San Diego. I have been able to observe 3 different classes taught at High Tech High during my Math Methods course. Understanding the history of High Tech High highlights some of the decisions and choices I saw them making while giving instruction there. It also helps to better understand the hiring practices and process that they go through for new teachers.

After reading this chapter I can clearly see why Wagner considers this to be a school that is working. Their focus on project based learning and assessment aligns with the goals and suggested practices that Wagner supports throughout this book. They are creating a curriculum at High Tech High that aligns with the needs of modern students and has been proven to prepare students for world outside of school.

I was helping a student in AVID with calculus homework when I confronted with the following problem.

I worked on it for probably 3 minutes before I called it and asked WolframAlpha for help. The response

Once I saw the arctan I remembered that this was a trig identity listed in the back of the book and that you needed to be memorize it. In the 2 years since Calc 3 I haven't used it so I forgot this one.

Since I have more mathematical knowledge at my disposile now than I did in during Calc 3 I thought I would dig around for a bit and try to remember why this specific identity related to the unit circle.

What I found was the Witch of Angensi. Having seen the curve and animation on the wiki page I then decided to try and recreate this model with Desmos.

In the end I was able to create a decent model of the Witch of Angensi and I thought I would share the process.

To start with I created the fixed lines and curves.

Next I created a variable t that would trace the function as it went from negative infinity to positive infinity. I also created a dashed vertical bar that would go along with the move.

Next I had to create a line from the origin to the boundary of y=1. I used the point-slope formula for this equation.

To have the line only extend from the origin to the point (t,1) I have to add bounds to the equation using { } brackets.

Next I had to write an equation for the point of intersection between the circle and the line.

This was actually harder than I thought it would be so I decided to start with a point on the circle and then fill in the rest.

So Starting over again.

We begin by creating an equation for the point on the circle as it moves from -pi to pi.

Next we again use the point slope formula to create an equation from the origin to the point on the circle and we create second equation for the line that extends from the circle to the boundary at y=1.

Using the x_2 we found in the last step we can create the other lines for the triangle.

You can now hit play on t and have a nicely animated version of the Witch of Agnesi.

If you want to expand your knowledge you can extend this model to include different values for the radius like I did in my example at the beginning of this post.

Like the last problem I find it's informative to start out with an inelegant solution and try to find an elegant one at the end.

We start by generating an array of the form a^b from a=2 to 100 and b=2 to 100.

As expected that got gigantic quickly, however we don't need to look at all of the data to work with it.

We can use Flatten to compress the 99 by 99 array into a single vector 9801 elements long.We then use DeleteDuplicates to remove all of the duplicated elements from the vector.And finish with Length to find the total number of elements left over.

And technically we are done. The answer to this ProjectEular problem is 9183.

However I think that it's worth while to consider why their are only 618 repeated numbers in this list.

First lets try to account for all of the numbers that are unique.

There are 25 prime numbers less than 100 and each of these numbers will generate 99 non-duplicated numbers since they will have only one number as their factor. So we have to have at least 2475 unique numbers just from the primes under 100.

However thinking about it we can also remove the numbers that are composed of primes like 6 and 15. Observe that since 6=2*3 that 6^2=(2*3)^2=2^2*3^3, and so on.

We could try to find the possible permutations of the 25 primes however there are 2^25 ways that we can combine the 25 primes and calculating which combinations have values less than 100 is rather time consuming.

Instead lets consider what numbers we know are going to show up more than once in the table?

Any number that is a power of a lower number of the list will definitely be a repeated.

The higher powers of 2 will show up in the powers of 4, 8, 16, 32, and 64.

Below I have generated a truncated list for the 2,4, and 8 table. We can see that the 4 table repeats every second value on the 2 table and that the 8 table repeats the 3rd value. This corresponds to 4=2^2 and 8=2^3.

I created the following two tables in Latex to highlight how the powers of 2 repeat at specific intervals in the 4 and 8 power tables. A similar pattern will occur for the powers of 3, 5, and 7.

Similarly the power table for 6 will be repeated in the power table for 36 and 10 in the power table for 100.

The following Matrix and ArrayPlot were generated in Mathematica to show the repeated numbers in the table given above. Observe that 2^6 occurs the most often in this table and is represented in black in the ArrayPlot to signify that it is a larger value.

Expanding the array to include all of the numbers we originally calculated we get the following ArrayPlot. Here we see that most of the values that will have repeats are clustered around the top of the array corresponding to a values between 2 and 10.

The dark band that shows up around the 2/3 mark on the array corresponds to the powers of 2^6. As you can see by the following table the values in the 2^6 power table show up in repeatedly throughout the array.

For reference the number that shows up the most often in the table is 1152921504606846976 or 2^60. Which makes sense when you remember that 60 has 12 divisors and would thus it will show up in the power table of 2^1, 2^2, 2^3, 2^4, 2^5, 2^6.

Also you should notice that in a number of rows the values are high for half of the row and then drop down to 1 for the rest of the array. As an example the power table for 100 is at the bottom of the array window. Note that the first 50 values of this table occur in the 10 power table.

It's not until 100^51 that we get the first value that doesn't show up in 10 power table.

Conclusion

This problem is mechanically simple to calculate with a computer but is calculable by hand given time and planning. I wouldn't ask students to do the full table by hand but I feel that you could probably have an interesting discussion with students concerning this material.

where |n| is the modulus/absolute value of n e.g. |11| = 11 and |−4| = 4

Find the product of the coefficients, a and b, for the quadratic expression that produces the maximum number of primes for consecutive values of n, starting with n = 0.

Initially I planned to tackle this problem from a strictly brute force perspective. I have found that working on a simple solution to a problem usually gives me insights to more elegant solutions.

Therefore I figured that the most straight forward and inelegant method was to produce a 1999 by 1999 array with individual elements of the array corresponding to the maximum number of consecutive primes generated by that combination of a and b.

Thus the initial element of the array in position |1,1| would be the maximum number of consecutive primes generated by n² + (-999)n + (-999), and the final position of |1999,1999| would be n² + (999)n + (999).

With this plan in mind I tried to think up a quick way to calculate the maximum number of consecutive primes for a given a and b.

I started by having Mathematica generate the first 101 values from the example given in the problem by using a=1 and b=41.

I then used PrimeQ to test whether each value was prime.

Then used Boole to convert these true false values to 0's and 1's.

And finally FirstPosition to find the first false in the list.

Note that the answer of 41 corresponds to the 39 primes listed in the example since this method doesn't count the primes from 0-39 but from 1-40 and it found a non-Prime number in position 41.

So...

This method is horribly inelegant, however it also does work so it's time to see if I can use this method to generate the 1999 by 1999 array I initially planned for.

I begin by trying a smaller array and seeing if my method works.

Note that this entry should be much larger as it displays a 51 by 51 array of values.

To find the maximum element of this array I then use the Position and Max commands. Position gives the location in an array of a specified value, and Max gives the largest element of the array.

Since we are counting up from 0 we have now found that the Maximum number of consecutive primes is generated with a=1 and b=41 which matches the example.

One area of concern going forward is that it takes 0.655204 secs for Mathematica to create the 51 by 51 array I used above. Since a 1999 by 1999 array is nearly 1600% larger than a 50 by 50 array that means that generating a 1999 by 1999 array should take at least 17 minutes and 28.33 second. Most likely longer considering that the edges of the array will be dealing with a and b values greater than 900.

Still I have free time and I'm not paying for processing time so we might as well go forward and see how long it takes.

So 21 minutes and 14 seconds later we have our array and can try to find the Maximum number of consecutive primes.

And we have a max at |939,1971|, so 939-1000=-61 and 1971-1000=971 are are a and b values.

a*b=-61*971=-59231

And we thus have our answer for Problem 27.

Now how could we have improved this process.

We could start by recognizing that b has to be a prime number. It was only after I had finished this problem that I realized that since n starts at 0 the first element will always be 0*0 + a*0 + b. Since there are 168 primes less than 1000 that means I only needed 168 rows instead of 1999.

A 1999 by 168 array is 8.4% the size of a 1999 by 1999 array and only took Mathematica 1 minute and 56 seconds to calculate.

A larger time sink is that I am actually calculating out the first 100 n values for every combination of a and b tested. Thus even if the second value calculated is non-Prime I still waste time evaluating the next 98 values.

Using the following code I can just check if the n values are prime and stop when a non-Prime is found.

To better understand how much time this saves, I calculated a 50 by 50 array using the old method and the new method and had Mathematica generate an ArrayPlot to save space.

The second method took 6.1% of the time taken by the first method.

Using both methods together the entire process of generating the 1999 by 168 array takes 10 seconds.

I haven't looked into it, but I suspect that I could similarly shave a couple of seconds off of the total running time by finding some trick for picking a values.

Conclusion

I started by creating a rather inelegant method for finding the number of consecutive primes generated on a 1999 by 1999 matrix. This method took 21 minutes and 14 seconds.

I next examined the problem more closely and found areas that could be refined so that I found that same solution in 10 seconds.

I'm confident that the majority of the improvement in performance came from optimizing the operation that found the number of consecutive primes.

Using the above code I found that 280607 times the second number tested was non-Prime. Since there are only 336000 entries in the array that means that 83.5% of the time while generating the original array I could have stopped after calculating the second value.

This was a longer write up than I originally expected, but it was a rather involved problem that benefited greatly from optimization.